from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5328, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,0,8]))
pari: [g,chi] = znchar(Mod(1477,5328))
Basic properties
| Modulus: | \(5328\) | |
| Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{592}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5328.lq
\(\chi_{5328}(181,\cdot)\) \(\chi_{5328}(1045,\cdot)\) \(\chi_{5328}(1117,\cdot)\) \(\chi_{5328}(1477,\cdot)\) \(\chi_{5328}(2125,\cdot)\) \(\chi_{5328}(2269,\cdot)\) \(\chi_{5328}(2845,\cdot)\) \(\chi_{5328}(3709,\cdot)\) \(\chi_{5328}(3781,\cdot)\) \(\chi_{5328}(4141,\cdot)\) \(\chi_{5328}(4789,\cdot)\) \(\chi_{5328}(4933,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1999,1333,2369,1297)\) → \((1,i,1,e\left(\frac{2}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5328 }(1477, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)